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Metamath Proof Explorer

Theorem cdlemefs44

Description: Value of f_s(r) when r is an atom under pq and s is any atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefs45 instead TODO: FIX COMMENT. (Contributed by NM, 31-Mar-2013)

Ref Expression
Hypotheses cdlemef44.b 
|- B = ( Base ` K )
cdlemef44.l 
|- .<_ = ( le ` K )
cdlemef44.j 
|- .\/ = ( join ` K )
cdlemef44.m 
|- ./\ = ( meet ` K )
cdlemef44.a 
|- A = ( Atoms ` K )
cdlemef44.h 
|- H = ( LHyp ` K )
cdlemef44.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemef44.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemef44.o 
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) )
cdlemef44.f 
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
cdlemefs44.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemefs44.i 
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
Assertion cdlemefs44 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = [_ R / s ]_ [_ S / t ]_ E )

Proof

Step Hyp Ref Expression
1 cdlemef44.b 
 |-  B = ( Base ` K )
2 cdlemef44.l 
 |-  .<_ = ( le ` K )
3 cdlemef44.j 
 |-  .\/ = ( join ` K )
4 cdlemef44.m 
 |-  ./\ = ( meet ` K )
5 cdlemef44.a 
 |-  A = ( Atoms ` K )
6 cdlemef44.h 
 |-  H = ( LHyp ` K )
7 cdlemef44.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemef44.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemef44.o 
 |-  O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) )
10 cdlemef44.f 
 |-  F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
11 cdlemefs44.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
12 cdlemefs44.i 
 |-  I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
13 eqid 
 |-  if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) = if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D )
14 eqid 
 |-  ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
15 eqid 
 |-  ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) )
16 1 2 3 4 5 6 7 8 11 12 13 9 10 14 15 cdlemefs31fv1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )
17 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A )
18 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
19 8 11 14 15 cdleme31sde 
 |-  ( ( R e. A /\ S e. A ) -> [_ R / s ]_ [_ S / t ]_ E = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )
20 17 18 19 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ [_ S / t ]_ E = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )
21 16 20 eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = [_ R / s ]_ [_ S / t ]_ E )